Even and odd functions: Difference between revisions

Revision as of 03:37, 14 April 2004

Revision as of 17:03, 15 April 2004

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Geometrically, an even function is [[symmetry|symmetric]] with respect to the ''y''-axis.

The designation '''even''' is due to the fact that the [[Taylor series]] of an even function includes only even powers and the [[Fourier series]] of an even function includes only [[trigonometric function|cosine]] terms.

Examples of even functions are ''x''², ''x''⁴, [[trigonometric function|cos]](''x''), and [[hyperbolic function|cosh]](''x'').

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Geometrically, an odd function is symmetric with respect to the [[origin]].

The designation '''odd''' is due to the fact that the Taylor series of an odd function includes only odd powers and the Fourier series of an odd function includes only [[trigonometric function|sine]] terms.

Examples of odd functions are ''x'', ''x''³, [[trigonometric function|sin]](''x''), and [[hyperbolic function|sinh]](''x'').

==Some facts==

===Basic properties===

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* In general, the [[addition|sum]] of an even and odd function is neither even nor odd; e.g. ''x'' + ''x''².

* The sum of 2 even functions is even, and any constant multiple of an even function is even. Also, The sum of 2 odd functions is odd, and any constant multiple of an odd function is odd.

* The [[multiplication|product]] of 2 even functions is an even function;.
* however, theThe product of 2 odd functions is alsoagain an '''even''' function.

* The [[derivative]] of an even function is odd,.
* and theThe derivative of an odd function is even.

===Series===

* The [[Taylor series]] of an even function includes only even powers.

* The [[Taylor series]] of an odd function includes only odd powers.

* The [[Fourier series]] of an even function includes only [[trigonometric function|cosine]] terms.

* The [[Fourier series]] of an odd function includes only [[trigonometric function|sine]] terms.

===Algebraic Structure===